On Numerical Solutions of Two-Dimensional Boussinesq Equations by Using Adomian Decomposition and He's Homotopy Perturbation Method

Transcription

1 Available a hp://pvam.ed/aam Appl. Appl. Mah. ISSN: Special Isse No. (Ags ) pp. Applicaios ad Applied Mahemaics: A Ieraioal Joral (AAM) O Nmerical Solios of Two-Dimesioal Bossiesq Eqaios by Usig Adomia Decomposiio ad He's Homoopy Perrbaio Mehod Syed Taseef Mohyd-Di Deparme of Mahemaics COMSATS Isie of Iformaio Techology Islamabad Pakisa syedaseefs@homail.com Msafa Ic ad Ebr Cavlak Deparme of Mahemaics Fıra Uiversiy 39 Elazı g Türkiye mic@fira.ed.r; ebrcavlak@homail.com.r Received: December 7 9; Acceped: Jly 5 Absrac I his paper we obai he approximae solio for -dimesioal Bossiesq eqaio wih iiial codiio by Adomia's decomposiio ad homoopy perrbaio mehods ad merical resls are compared wih exac solios. Keywords: Bossiesq eqaios Adomia s polyomials oliear problems homoopy perrbaio error esimaes MSC #: 65 N. Irodcio Oe of he eqaio describig he propagaio of log waves o shallow waer is he Bossiesq oe which firs appeared. The ahor was he firs o explai scieifically he effec of he exisece of soliary waves or solios discovered i 834 by Sco-Rssell. Bossiesq eqaio ca be wrie i he form

2 Mohyd-Di e al. = ( ) where ( x is a elevaio of he free srface of flid ad he coefficies = cos R. The Bossiesq eqaio was proposed earlier ha Koreweg-de Vries oe b he mahemaical heory for i is o as complee as for he laer oe. The oe-dimesioal i space Bossiesq eqaio ad is geeralizaio = ( f ( )) have bee sdied i may papers. The wo-dimesioal versio of he geeralized Bossiesq eqaio yy m m = = has bee proposed by Ablowiz e al. (997). This eqaio is moivaed by cosideraios derlyig he derivaio of he Kadomsev-Peviashvili ype eqaios ad models slow rasverse variaios balaced by logidial dispersio ad weak olieariy. I his sdy we also cosider he -dimesioal Bossiesq eqaio ) = () 3( sbjec o iiial codiios c c ( x) = sec h ( x ). (). The Adomia Decomposiio Mehod (ADM) The ADM was firs irodced by Adomia i he begiig of 98's. The mehod is sefl obaiig boh a closed form ad he explici solio ad merical approximaios of liear or oliear differeial eqaios ad i is also qie sraighforward o wrie comper codes. This mehod has bee applied o obai a formal solio o a wide class of sochasic ad deermiisic problems i sciece ad egieerig ivolvig algebraic differeial iegrodiffereial differeial delay iegral ad parial differeial eqaios Lesic e al. (999) ad Dehgha (4). The covergece of ADM for parial differeial eqaios was preseed by Cherral (99). Applicaio ad covergece of his mehod for oliear parial differeial eqaios are fod i Ngarhasa e al. () ad Hashim e al. (6).

3 AAM: Ier. J. Special Isse No. (Ags ) 3 I geeral i is ecessary o corc he solio of he problems i he form of a decomposiio series solio. I he simples case he solio ca be developed as a Taylor series expasio abo he fcio o he poi a which he iiial codiio ad iegraio righ had side fcio of he problem are deermied he firs erm of he decomposiio series for. The sm of he erms are simply he decomposiio series Adomia (989) Adomai (994) Adomai (998) ad Dehgha (4). x = x. = (3) Sppose ha he differeial eqaio operaor icldig boh liear ad oliear erms ca be formed as L R N = Fx (4) wih iiial codiio x = gx (5) where L is he higher-order derivaive which is assmed o be iverible R is a liear differeial operaor of order less ha F x is a sorce erm. We ex apply he iverse operaor codiio (5) o obai L N is he oliear erm ad L o boh sides of eqaio (4) ad sig he give x = gx f x L R L N (6) where he fcio f x represes he erms arisig from iegraig he sorce erm F x ad from sig he give codiios all are assmed o be prescribed. The oliear erm ca be wrie as El-Sayed () Ic (6) ad Ic (7). N = = A (7) where A are he Adomia polyomials. These polyomials are defied as A = d k N! d = k x =. k= (8) For example A N =

4 4 Mohyd-Di e al. ' A = N ' '' A = N N ' '' A 3N N 6 ''' 3 = 3 N (9) ad so o he oher polyomials ca be cosrced i a similar way Wazwaz (). As idicaed before Adomia mehod defies he solio by a ifiie series of compoes give by eqaio (4) ad he compoes are sally recrrely deermied. Ths he formal recrsive relaio is defied by x = g x f x x = L R L N () which are obaied all compoes of. As a resl he erms of he series are ideified ad he exac solio may be eirely deermied by sig he approximaio x = lim x () where x = x k k = () or = = = =. (3) Eqaio () ca be rewrie i a operaor form L L 3( ) L = (4) x where he liear differeial operaors L L x ad respecively. Assmig he iverse of he operaor L are give by / / x 4 4 ad / x L exiss ad i ca coveiely be ake as

5 AAM: Ier. J. Special Isse No. (Ags ) 5 he defiie iegral wih respec o from o ha is L =. dd. mehod sggess ha he kow fcios be decomposed by a ifiie series The decomposiio x = x = (5) A x = ( ) = = ad he oliear erms polyomials ad hese polyomials ca be calclaed as A. I here A are he so-called Adomia A = 3( ) A = 3( ) A = 3( ) (6) Ths applyig he iverse operaor L o (4) yields L L = L L 3( ) L. x (7) Therefore eqaios () are rasformed io a se of recrsive relaios give by ( x = x x x L L A L ( ) = ( ) x = L Lxk Ak Lk k (8) 3. The Homoopy Perrbaio Mehod The homoopy perrbaio mehod (HPM) was firs proposed by He (998) He (4) He (4) He (4) He (5) He (5) He (6). The HPM does o deped o a small parameer i he eqaio. Usig homoopy echiqe i opology a homoopy is cosrced p which is cosidered as a " small parameer". wih a embeddig parameer The HPM was sccessflly applied o oliear oscillaors wih discoiiies He (4) ad bifrcaio of oliear problems He (4). I He (4) a compariso of HPM ad homoopy aalysis mehod was made revealig ha he former is more powerfl ha he laer. The HPM was proposed o search for limi cycles or bifrcaio crves of oliear eqaios He (5). I He (5) herisical example was give o illsrae he basic idea of he HPM. Also his mehod was applied o solve bodary vale problems He (6) ad hea radiaio eqaios Gaji e al. (6) ad Noor e al.

6 6 Mohyd-Di e al. Whe implemeig he HPM we ge he explici solios of he wo-dimesioal parabolic eqaio wiho sig ay rasformaio mehod. The mehod preseed here is also simple o se for obaiig merical solio of he eqaios wiho sig ay discreizaio echiqes. Frhermore we will show ha cosiderably beer approximaios relaed o he accracy level ca be obaied. To illsrae he basic ideas of his mehod we cosider he followig oliear differeial eqaio: A f r = r (9) wih he bodary codiios of B / = r () f r a kow aalyical where A is a geeral differeial operaor B a bodary operaor fcio ad is he bodary of he domai ad o deoes differeiaio alog he ormal vecor draw owards from. Geerally speakig he operaor A ca be divided io wo pars which are L ad N where L is liear b N is oliear. Eqaio () ca herefore be rewrie as follows: L N f r =. () By he homoopy echiqe we cosrc a homoopy H : ] R which saisfies: H V p = p [ L V r L r ] p[ A V r f r ] = p r () or H V p = L V r L r p L r p[ N V r f r ] = (3) where p is a embeddig parameer is a iiial approximaio of eqaio () which saisfies he bodary codiios. Obviosly from eqaios () ad () we will have: V = LV r L r= H (4) V = AV r f r =. H (5) Chagig process of p from zero o iy is js ha V r p chages from r o r opology his is called deformaio ad L V r L r ad AV r f r homoopy.. I are called

7 AAM: Ier. J. Special Isse No. (Ags ) 7 Accordig o he HPM we ca firs se he embeddig parameer p as a " small parameer" ad assme ha he solio of eqaio (3) ad (4) ca be wrie as a power series i p: V = V pv p V. (6) Seig p = resls i he approximae solio of eqaio (): = lim V = V V V p. (7) The combiaio of he perrbaio mehod ad he homoopy mehod is called he homoopy perrbaio mehod (HPM) which has elimiaed he limiaios of he radiioal perrbaio mehods. O he oher had his echiqe ca have fll advaage of he radiioal perrbaio echiqes. The series (6) is coverge for mos cases He ( ). To ivesigae he ravelig wave solio of eqaio () we firs cosrc a homoopy as follows: '' '' ( IV ) ( p)( Y ) p( Y Y 3( Y ) Y ) = (8) where " primes" deoe differeiaio wih respec o x ad " do" deoes differeiaio wih respec o. Sbsiig eqaio (4) ad arragig he coefficies of p powers we have Y 3 4 py py p3y p4y p '' '' 3 '' 4 '' 5 '' py p Y p Y p Y3 p Y4 3p Y 6p Y Y 3pY 6p Y Y 6p Y Y (3) '' '' 4 '' '' 3 '' '' 5 '' 5 '' '' 4 '' '' 5 '' '' 3p Y 6p Y Y3 6p Y Y3 6p Y Y4 ( IV ) ( IV ) 3 ( IV ) 4 '( IV ) 5 '( IV ) py p Y p Y p Y3 p Y4 =. I order o obai he kowsof Y i ( x i = 34 we ms cosrc ad solve he followig sysem which icldes five eqaios wih five kows cosiderig he iiial codiio of Y ( x)= ( x) ad havig he iiial approximaios of eqaio (): p Y = '' '' ' Y Y 3Y Y = '' '' '' '' ' Y Y 3Y 6Y Y Y = : v : p p v : p : Y Y 6 Y Y Y = p p 3 '' '' '' ( IV ) 3 4 '' '' '' '' '' ' v : Y 4 Y3 6Y Y 6Y Y3 Y3 = 5 '' '' '' '' '' '' ' v : Y 5 Y4 3Y 6Y Y3 6Y Y4 Y4 = (4)

8 8 Mohyd-Di e al. 4. Tes he Example I his secio we prese he -dimesioal Bossiesq eqaio wih aalyical solios o show he efficiecy of mehods described i he previos secio. We shall cosider eqaio () wih he followig iiial codiios. These gives he exac solio c c c x = sech x c. Firs we apply he ADM o eqaio (). To cosrc he correcio fcioal i is sfficie o se Eqs.(6) ad (8). c c = sec h ( x ) c c c h x h x 4 5/ = sec ( ) a ( )... c c h x c c c x c c x 8 4 = sec ( ) ( ( ) ( ) cosh[ ( )] sih[ ( )]) c 3 = ( c ( c) sec h ( x ) (4 44c 5(33 c) cosh[ c( x )] 536 4(3c ) cosh[ c( x )] cosh[3 c( x )] ccosh[3 c( x )] (35) ad so o i his maer he oher compoes of he decomposiio series (5) were obaied of which was evalaed o have he followig expasios: x = 3 = sech c 4 c ( x ) ( c) c ( cosh[ c( x )]sech ( x ) (( c) 8 c ( 4 57 cosh[ cosh[3 c( x )]sech c( x )]) 6 cosh[ c ( x ). c( x )]

9 AAM: Ier. J. Special Isse No. (Ags ) 9 We ow apply he HPM o eqaio () we obaied i sccessio eqaio (3) as 3 ec. by sig c c Y = sech ( x ) 4 c Y = ( c) c ( cosh[ c( x )]sech ( x ) Y = (( c) c ( c 948 c 336 c 336 c 37 3 (5 385c 884 c 6c ( Y3 = (( c) c ( c c c c c ad so o i he same maer he oher compoes ca be obaied sig he Mahemaica package. Table. Error bewee he ADM sig 6 erms ad exac solios of ( x for c =. /x ¹¹ ¹ ¹ ¹ ² ¹¹ ¹³ ¹ ¹ ¹ ¹¹¹ ¹³ ¹ ¹ ¹ ¹ ¹² ¹ ¹ ¹ ¹ ¹² ¹ ¹ ¹ ¹ ¹² ¹ ¹ ¹ ¹ ¹² ¹ ¹ ¹ ¹ ¹² ¹ ¹ ¹ ¹ ¹² ¹ ¹ ¹² ¹ ¹ ¹ Table. Error bewee he ADM sig 6 erms ad exac solios of ( x for c =. /x ¹⁵ ¹⁸ ²¹ ²⁴ ²⁷ ¹⁴ ¹⁷ -.58 ²⁰ ²³ ²⁶ ¹⁴ ¹⁷ ²⁰ ²³ ²⁶ ¹³ -.48 ¹⁶ ²⁰ ²³ ²⁶ ¹³ ¹⁶ ¹⁹ -.34 ²² ²⁵ ¹³ ¹⁶ ¹⁹ ²² ²⁵ ¹³ ¹⁶ ¹⁹ ²² ²⁵ ¹³ ¹⁶ ¹⁹ ²² ²⁵ ¹³ ¹⁶ ¹⁹ ²² ²⁵ ¹⁶ ¹⁶ ¹⁹ ²² ²⁵

10 Mohyd-Di e al. Table 3. Error bewee he HPM sig 6 erms ad exac solios of ( x for c =. /x ¹¹ ¹³ ¹⁵ ¹⁷ ¹⁹ ¹⁰ ¹³ ¹⁵ -4.3 ¹⁷ ¹⁹ ¹⁰ ¹² -9.5 ¹⁵ ¹⁷ ¹⁹ ¹⁰ ¹² -.56 ¹⁴ ¹⁷ ¹⁹ ¹⁰ ¹² ¹⁴ ¹⁷ ¹⁹ ¹⁰ ¹² -.63 ¹⁴ ¹⁶ ¹⁹ ¹⁰ ¹² -.84 ¹⁴ ¹⁶ ¹⁹ ¹⁰ ¹² ¹⁴ ¹⁶ ¹⁹ ¹⁰ -3.9 ¹² ¹⁴ ¹⁶ ¹⁹ ¹⁰ ¹² ¹⁴ ¹⁶ -.67 ¹⁹ Table 4. Error bewee he HPM sig 6 erms ad exac solios of ( x for c =. /x ¹⁴ ¹⁷ ²⁰ ²³ ²⁶ ¹⁴ ¹⁷ ²⁰ ²³ ²⁶ ¹⁴ ¹⁷ ²⁰ ²³ ²⁶ ¹⁴ ¹⁷ -8.3 ²⁰ ²³ ²⁶ ¹⁴ ¹⁷ ²⁰ ²³ ²⁶ ¹⁴ ¹⁷ ²⁰ ²³ ²⁶ ¹⁴ ¹⁷ ²⁰ ²³ ²⁶ ¹⁵ ¹⁷ ²⁰ ²³ ²⁶ ¹⁴ ¹⁷ ²⁰.866 ²³.496 ²⁶..79 ¹³ ¹⁷ 7.77 ²⁰ ²³ ²⁶ 5. Coclsio Ths we have illsraed how Adomia decomposiio mehod ad homoopy perrbaio mehod ca be sed o solve of Bossiesq eqaio. The accracy of he merical solios ivesigaed ha he mehods is well sied for he solio of he oliear eqaios. The resls of merical example is preseed ad oly few erms are reqired o obai accrae solios. I Table ad shows ha he error bewee he exac vale of ad he approximaio of. The errors obaied by sig he approximae solio give i sig oly few erms ieraios of he decomposiio mehod. I Table 3 ad 4 show ha he exac ad merical solios are for oly few erms by sig Homoopy perrbaio mehod. Refereces Ablowiz M. J. ad Wag X. P. (997). Sd. Appl. Mah. Vol. 98 pp Adomia G. (989). Nol. Sochasic Sysems Appl. o Physics Klwer Academic Press Boso MA.

11 AAM: Ier. J. Special Isse No. (Ags ) Adomia G. (994). Solvig Froier Problems of Physics: The Decomposiio Mehod Klwer Academic Press Boso MA. Adomia G. (998). J. Mah. Aal. Appl. Vol. 35 pp Cherral Y. (99). Mah. Comp. Modellig Vol. 4 pp Dehgha M. (4). Appl. Mah. Comp. Vol. 57 pp Dehgha M. (4). I. J. Comper Mah. Vol. 8 pp El-Sayed M.S. (). Appl. Mah. Comp. Vol. 3 pp Gaji D.D. ad Rajabi A. (6). I. Comm. Hea Mass Trasfer Vol. 33 pp. 39. Hashim I. Noorai M. S. ad Al-Hadidi M. R. S. (6). Mah. Comp. Modellig Vol. 43 pp He J. H. (998). Comm. Noliear Sci. Nmer. Siml. Vol. 3 () pp.6. He J. H. (4). Appl. Mah. Comp. Vol. 56 pp. 57. He J. H. (4). Appl. Mah. Comp. Vol. 5 pp. 87. He J. H. (4). Appl. Mah. Comp. Vol. 56 (3) pp. 59. He J. H. (5). I. J. Noliear Sci. Nmer. Siml. Vol. 6 () pp.7. He J. H. (5). Chaos Solios Fracals Vol. 6 pp He J. H. (6). Phys. Le. A Vol. 35 pp. 87. Ic M. (6). Appl. Mah. Comp. Vol. 73 pp Ic M. (7). Chaos Solios Fracals Vol. 33 pp Lesic D. ad Ellio L. (999). J. Mah. Aal. Appl. Vol. 3 pp Ngarhasa N. Some B. Abbaoi K. Cherral Y. (). Kyberees Vol. 3 pp Noor M. A. ad Mohyd-Di S. T. (8). Comp. Mah. Appl. Vol. 55 () pp Wazwaz A.M. (). Parial Differeial Eqaios: Mehods ad Applicaios The Neherlads Balkema.